I have come to like the ams bibliography style. Is there a rationale for the style available? In general, are there guidelines for typesetting bibliographies you can recommend? Lastly, what are your favourite bibliography styles?

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The AMS bibliography style is based on tradition. Since it really hasn't changed for as long as I can remember (or check), it's likely that no one is available who can answer your first question with authority.

If you like this style, it is described in Ellen Swanson's Mathematics into Type, which was first published in 1971. Ellen was Director of Editorial Services at the AMS from the early 1950s until her retirement in the mid 1980s.

It is my understanding that the updated edition (1999) of this book, now available only by purchase (or from a library) has been cleared for posting on line as a PDF file, to be freely available; however, no date has been set for its release.

Another style manual often suggested for authors publishing in the U.S. is the Chicago Manual of Style. A "citation quick guide" is freely available on line.

Edit: A pdf file of Mathematics into Type has been posted for on-line access. It's still under copyright; redistribution is discouraged, but linking is okay. The terms of access are linked from a note at the bottom of every page of the manual.